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An embedded (resp. immersed) degenerate submanifold of a proper
pseudo-Riemannian manifold\((M,g)\) is an embedded (resp. immersed)
submanifold \(H\) of \(M\) as a differentiable manifold such that pull
back of the metric tensor \(g\) via the embedding (resp. immersion)
endows \(H\) with the structure of a degenerate manifold.

Degenerate submanifolds are study in many fields of mathematics and physics,
for instance in Differential Geometry (especially in geometry of
lightlike submanifold) and in General Relativity. In geometry of lightlike
submanifolds, according to the dimension \(r\) of the radical distribution
(see below for definition of radical distribution), degenerate submanifolds
have been classify into 4 subgroups: \(r\)-lightlike submanifolds, Coisotropic
submanifolds, Isotropic submanifolds and Totally lightlike submanifolds.
(See the book of Krishan L. Duggal and Aurel Bejancu [DS2010].)

In the present module, you can define any of the 4 types but most of the
methods are implemented only for degenerate hypersurfaces who belong to
\(r\)-lightlike submanifolds. However, they might be generalized to
\(1\)-lightlike submanifolds. In the literature there is a new approach
(the rigging technique) for studying \(1\)-lightlike submanifolds but
here we use the method of Krishan L. Duggal and Aurel Bejancu based on
the screen distribution.

Let \(H\) be a lightlike hypersurface of a pseudo-Riemannian manifold
\((M,g)\). Then the normal bundle \(TH^\perp\) intersect the tangent
bundle \(TH\). The radical distribution is defined as
\(Rad(TH)=TH\cap TH^\perp\). In case of hypersurfaces, and more
generally \(1\)-lightlike submanifolds, this is a rank 1 distribution.
A screen distribution \(S(TH)\) is a complementary of \(Rad(TH)\) in \(TH\).

Giving a screen distribution \(S(TH)\) and a null vector field \(\xi\)
locally defined and spanning \(Rad(TH)\), there exists a unique
transversal null vector field ‘N’ locally defined and such that
\(g(N,\xi)=1\). From a transverse vector ‘v’, the null rigging ‘N’
is giving by the formula

Tensors on the ambient manifold \(M\) are projected on \(H\) along \(N\)
to obtain induced objects. For instance, induced connection is the
linear connection defined on H through the Levi-Civitta connection of
\(g\) along \(N\).

To work on a degenerate submanifold, after defining \(H\) as an instance
of DifferentiableManifold,
with the keyword structure='degenerate_metric', you have to set a
transvervector \(v\) and screen distribution together with the radical
distribution.

An example of degenerate submanifold from General Relativity is the
horizon of the Schwarzschild black hole. Allow us to recall that
Schwarzschild black hole is the first non-trivial solution of Einstein’s
equations. It describes the metric inside a star of radius \(R = 2m\),
being \(m\) the inertial mass of the star. It can be seen as an open
ball in a Lorentzian manifold structure on \(\RR^4\):

An embedded (resp. immersed) degenerate submanifold of a proper
pseudo-Riemannian manifold\((M,g)\) is an embedded (resp. immersed)
submanifold \(H\) of \(M\) as a differentiable manifold such that pull
back of the metric tensor \(g\) via the embedding (resp. immersion)
endows \(H\) with the structure of a degenerate manifold.

INPUT:

n – positive integer; dimension of the manifold

name – string; name (symbol) given to the manifold

field – field \(K\) on which the manifold is
defined; allowed values are

'real' or an object of type RealField (e.g., RR) for
a manifold over \(\RR\)

'complex' or an object of type ComplexField (e.g., CC)
for a manifold over \(\CC\)

signature – (default: None) signature \(S\) of the metric as a
tuple: \(S = (n_+, n_-, n_0)\), where \(n_+\) (resp. \(n_-\), resp. \(n_0\)) is the
number of positive terms (resp. negative terms, resp. zero tems) in any
diagonal writing of the metric components; if signature is not
provided, \(S\) is set to \((ndim-1, 0, 1)\), being \(ndim\) the manifold’s dimension

ambient – (default: None) manifold of destination
of the immersion. If None, set to self

base_manifold – (default: None) if not None, must be a
topological manifold; the created object is then an open subset of
base_manifold

latex_name – (default: None) string; LaTeX symbol to
denote the manifold; if none are provided, it is set to name

start_index – (default: 0) integer; lower value of the range of
indices used for “indexed objects” on the manifold, e.g., coordinates
in a chart
- category – (default: None) to specify the category; if
None, Manifolds(field) is assumed (see the category
Manifolds)

unique_tag – (default: None) tag used to force the construction
of a new object when all the other arguments have been used previously
(without unique_tag, the
UniqueRepresentation
behavior inherited from
ManifoldSubset
would return the previously constructed object corresponding to these
arguments)

This method is implemented only for null hypersurfaces. The method
returns a tensor \(B\) of type \((0,2)\) instance of
TangentTensor
such that for two vector fields \(U, V\) on the ambient manifold along
the null hypersurface, one has:

\[\nabla_UV=D(U,V)+B(U,V)N\]

being \(\nabla\) the ambient connection, \(D\) the induced connection
and \(N\) the chosen rigging.

INPUT:

screen – (default: None) an instance of
Screen.
If None, the default screen is used

Return the restriction of the ambient metric on vector field
along the submanifold and tangent to it. It is difference from
induced metric who gives the pullback of the ambient metric
on the submanifold.

For a given tensor \(T\) of type \((r, 1)\) on the ambient manifold, this
method returns the tensor \(T'\) of type \((r,1)\) such that for \(r\)
vector fields \(v_1,\ldots,v_r\), \(T'(v_1,\ldots,v_r)\) is the projection
of \(T(v_1,\ldots,v_r)\) on self along the bundle spanned by the
transversal vector fields provided by set_transverse().

For setting a screen distribution and vector fields of the radical distribution
that will be used for computations

INPUT:

name – string (default: None); name given to the screen

latex_name – string (default: None); LaTeX symbol to denote
the screen; if None, the LaTeX symbol is set to name

screen – list or tuple of vector fields
of the ambient manifold or chart function; of the ambient manifold in
the latter case, the corresponding gradient vector field with respect to
the ambient metric is calculated; the vectors must be linearly independent,
tangent to the submanifold but not normal

rad – – list or tuple of vector fields
of the ambient manifold or chart function; of the ambient manifold in
the latter case, the corresponding gradient vector field with respect to
the ambient metric is calculated; the vectors must be linearly independent,
tangent and normal to the submanifold

For a given tensor \(T\) of type \((r, 1)\) on the ambient manifold, this
method returns the tensor \(T'\) of type \((r,1)\) such that for \(r\)
vector fields \(v_1,\ldots,v_r\), \(T'(v_1,\ldots,v_r)\) is the projection
of \(T(v_1,\ldots,v_r)\) on the bundle spanned by screen along the
bundle spanned by the transversal plus the radical vector fields provided.

This method is implemented only for null hypersurfaces. The method
returns a tensor \(B\) of type \((0,2)\) instance of
TangentTensor
such that for two vector fields \(U, V\) on the ambient manifold along
the null hypersurface, one has:

\[\nabla_UV=D(U,V)+B(U,V)N\]

being \(\nabla\) the ambient connection, \(D\) the induced connection
and \(N\) the chosen rigging.

INPUT:

screen – (default: None) an instance of
Screen.
If None, the default screen is used

For setting a transversal disttribution of the degenerate submanifold.
according to the type of the submanifold amoung the 4 possible types,
one must enter a list of normal transversal vector fields and/or a
list of transversal and not normal vector fields spanning a transverse
distribution.

INPUT:

rigging – list or tuple (default: None); list of vector fields
of the ambient manifold or chart function; of the ambient manifold in
the latter case, the corresponding gradient vector field with respect to
the ambient metric is calculated; the vectors must be linearly independent,
transversal to the submanifold but not normal

normal – list or tuple (default: None); list of vector fields
of the ambient manifold or chart function; of the ambient manifold in
the latter case, the corresponding gradient vector field with respect to
the ambient metric is calculated; the vectors must be linearly independent,
transversal and normal to the submanifold

Let \(H\) be a lightlike submanifold embedded in a pseudo-Riemannian
manifold \((M,g)\) with \(\Phi\) the embedding map. A screen distribution
is a complementary \(S(TH)\) of the radical distribution \(Rad(TM)=TH\cap
TH^\perp\) in \(TH\). One then has

Return either a list Rad of vector fields spanning the
complementary of the normal distribution \(TH^\perp\) in the
transverse bundle or (when \(H\) is a null hypersurface) the
null transversal vector field defined in [DB1996].

OUTPUT:

either a list made by vector fields or a vector field in
case of hypersurface